Complete the square
to find the vertex
of this parabola.

([?], [ ])

Question

Complete the square to find the vertex of this parabola. $$ x^{2}-14 x+4 y+29=0 $$ ([?], [ ])

UpStudy AI · Accepted Answer

**Step 1. Write the equation and complete the square in $$ x $$**

The given equation is
$$
x^{2} - 14x + 4y + 29 = 0.
$$
Focus on the $$ x $$-terms:
$$
x^{2} - 14x.
$$
To complete the square, add and subtract $$\left(\frac{-14}{2}\right)^2 = 49$$:
$$
x^{2} - 14x = \left(x^{2} - 14x + 49\right) - 49 = (x - 7)^2 - 49.
$$

**Step 2. Substitute the completed square back into the equation**

Replace $$ x^{2} - 14x $$ in the original equation:
$$
(x - 7)^2 - 49 + 4y + 29 = 0.
$$
Combine the constants:
$$
-49 + 29 = -20.
$$
Thus, the equation becomes:
$$
(x - 7)^2 + 4y - 20 = 0.
$$

**Step 3. Solve for $$ y $$ to obtain the vertex form**

Isolate the $$ y $$-term:
$$
4y = 20 - (x - 7)^2,
$$
$$
y = 5 - \frac{1}{4}(x - 7)^2.
$$

Now the equation is in the form:
$$
y = k - a(x - h)^2,
$$
where the vertex is $$(h, k)$$.

**Step 4. Identify the vertex**

From the equation, we see that:
$$
h = 7 \quad \text{and} \quad k = 5.
$$

Thus, the vertex of the parabola is $$(7, 5)$$.
The vertex of the parabola is at $$(7, 5)$$.

Complete the square
to find the vertex
of this parabola.

([?], [ ])

Upstudy AI Solution

Answer

Solution

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Complete the square to find the vertex of this parabola. ([?], [ ])

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Complete the square
to find the vertex
of this parabola.

([?], [ ])